Great circles and airline routes

2006-10-09

in Geek stuff, Science, Travel

When flying between western Canada and England, it sometimes seems surprising that such a northward trajectory is followed. On my way back to Vancouver, for instance, we were treated to an aerial view of Iceland’s unique landscape. Of course, the reason for the path is that the spherical character of the earth is not well reflected in standard map projections. The most famous – the Mercator projection – is arranged such that a straight line drawn on the map will correspond to a course that actually passes through each point on the earth depicted. This kind of map is called ‘conformal.’ As such, the notorious distortion (enlarging the apparent size of polar regions while reducing that of equatorial ones) is an emergent property of its design.

That said, the most efficient course between any two points on the globe is probably not the one that connects them on a Mercator projection line of the shortest distance. Mathematically, the most direct course is based on what is called a ‘great circle.’ That is to say, imagine marking your present location and your destination using a marker on an orange. The line you could draw all the way around, intersecting both, is the great circle. The line segment between the points is the shortest distance that can be transcribed between them on a sphere (or near-sphere, in the case of the earth).

Unless you are going due north, due south, or straight around the equator, actually following a great circle path requires constantly changing your heading. This is because of how the line you are on does not maintain a constant bearing with respect to either magnetic or true north. In the days before computers and long haul air travel, few people would probably have bothered to calculate great circle courses. A more venerable option can be found in the Rhumb line. Now, GPS and autopilot systems have made doing so all but automatic. Hence the genesis of those gracefully arcing lines printed in your in-flight magazine.

On a separate note, the precision of modern location and navigation systems in aircraft can sometimes cause problems. (Via Philip Greenspun)

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{ 2 comments… read them below or add one }

Milan February 28, 2008 at 11:06 am
. March 29, 2010 at 4:55 pm

“For example, when I was little, my dad used to enjoy quizzing me about geography. Which is farther north, he’d ask, Rome or New York City? Most people would guess New York, but surprisingly they’re at almost the same latitude, with Rome being just a bit farther north. On the usual map of the world (the misleading Mercator projection, where Greenland appears gigantic) it looks like you could go straight from New York to Rome by heading due east.

Yet airline pilots never take that route. They always fly northeast out of New York, hugging the coast of Canada. I used to think they were staying close to land for safety’s sake, but that’s not the reason. It’s simply the most direct route, if you take the earth’s curvature into account. The shortest path from New York to Rome goes past Nova Scotia and Newfoundland, then heads out over the Atlantic, and finally veers south of Ireland and across France for arrival in sunny Italy.

This kind of path on the globe is called an arc of a “great circle.” Like straight lines in ordinary space, great circles on a sphere contain the shortest paths between any two points. They’re called “great” because they’re the largest circles you can have on a sphere. Conspicuous examples include the equator and the longitudinal circles that pass through the north and south poles.

Another property that lines and great circles share is that they’re the straightest paths. That might sound strange — all paths on a globe are curved, so what do we mean by “straightest”? Well, some paths are more curved than others. The great circles don’t do any additional curving, above and beyond what they’re forced to do by following the surface of the sphere.”

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